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Error Bound for Conic Inequality

Abstract

In this note, we consider error bound issue for conic inequalities. In terms of subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In particular, our result extends Ioffe’s classical result on local error bounds to the vector-valued function case.

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References

  1. Azé, D., Corvellec, J.-N.: Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var. 10, 409–425 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions on metric spaces. Vietnam J. Math. 40, 165–180 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions: Necessary and sufficient conditions. Nonlinear Anal. 75, 1124–1140 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  5. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Nat. Bur. Stand. 49, 263–265 (1952)

    Article  Google Scholar 

  6. Ioffe, A.D.: Regular points of Lipschitz functions. Trans. Amer. Math. Soc. 251, 61–69 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surveys 55, 501–558 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lewis, A., Pang, J.S.: Error bounds for convex inequality systems. In: Crouzeix, J.-P., Martinez-Legaz, J.-E., Volle, M (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results, 75–100. Proceedings of the Fifth Symposium on Generalized Convexity, Luminy, June 1996. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  9. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I/II. Springer, Berlin, Heidelberg (2006)

    Google Scholar 

  10. Ng, K.F., Zheng, X.Y.: Error bound for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12, 1–17 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ngai, H.V., Théra, M.: Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math. Program. 116, 397–427 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pang, J.S.: Error bounds in mathematical programming. Math. Program. Ser. B 79, 299–332 (1997)

    MATH  Google Scholar 

  13. Robinson, S.M.: An application of error bounds for convex programming in a linear space. SIAM J. Control 13, 271–273 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ursescu, C.: Multifunctions with closed convex graph. Czech Math. J 25, 438–441 (1975)

    MathSciNet  Google Scholar 

  16. Wu, Z., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program. 92, 301–314 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zalinescu, C.: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces. In: Proc. 12th Baikal Internat. Conf. on Optimization Methods and Their Appl, pp. 272–284. Irkutsk, Russia (2001)

  18. Zheng, X.Y., Ng, K.F.: Error bound moduli for conic convex systems on Banach spaces. Math. Oper. Res. 29, 213–228 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zheng, X.Y., Ng, K.F.: Pertubation analysis of error bounds for systems of conic linear inequalities. SIAM J. Optim. 15, 1026–1041 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zheng, X.Y., Ng, K.F.: Metric subregularity and constraint qualifications for convex generalized equations in Banach spaces. SIAM J. Optim. 18, 437–460 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zheng, X.Y., Ng, K.F.: Metric subregularity and calmness for nonconvex generalized equations in Banach spaces. SIAM J. Optim. 20, 2119–2136 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Xi Yin Zheng.

Additional information

This paper is dedicated to Professor Boris Mordukhovich on the occasion of his 65th birthday.

The first author was supported by the National Natural Science Foundation of P.R. China (Grant No. 11371312) and IRTSTY. The second author was supported by General Research Fund grants from the Research Grant Council of Hong Kong (RGC Ref. Nos. CUHK 403110 and 402612).

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Zheng, X.Y., Ng, K.F. Error Bound for Conic Inequality. Vietnam J. Math. 42, 509–519 (2014). https://doi.org/10.1007/s10013-014-0098-7

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  • DOI: https://doi.org/10.1007/s10013-014-0098-7

Keywords

  • Conic inequality
  • Error bound
  • Subdifferential

Mathematics Subject Classification (2000)

  • 90C31
  • 90C25
  • 49J52